Econ 251a
Fall 2006
John Geanakoplos
Dynamically
Hedging Call Options
Abstract:
Hedging and Pricing are simultaneous operations.
1 Pricing
Consider a stock selling for $100. Over the course of a year it is expected to increase
in value by 10%, with a standard deviation of 16%. These are typical numbers for
reason the stock price is only $100, instead of $110/1.06, is the risk. In the states next
year when the stock price is well above $110, the economy as a whole is probably also
doing well, so the marginal investor is eating pretty well, and so his marginal utility
for money is relatively low. In the states when the stock price next year is well below
$110, the economy is probably doing relatively badly, and the typical consumer has a
high marginal utility of money. Thus the stock pays most when the money is needed
least, and so it is worth less than an asset which delivered $110 in every state.
Consider a call option on this stock with expiration date one year into the future,
with exercise price 106. (Note that the present value of the exercise price is equal
to the stock price, which is the case for which the call option has the same value as
the put option, and also is a special case where the BlackScholes formula for the call
option price is particularly simple). It can be shown that the expected payo
f
from
the call option is about $9.4(1.06). One might think therefore that the call option
should sell for $9.4. But that is wrong. The call option pays most when the stock
price is highest, which we have seen is when the marginal utility of money is lowest.
Hence the call option should sell for less than $9.4. But for how much less? One does
not seem to have enough information to answer this question.
Suppose that (1) the interest rate will never change, and that (2) the stock price
moves continuously through time, and that (3) the percentage changes in the stock
price are identically distributed over equally long intervals, and that (4) the percent
age change in stock price over any time interval is independent of the percentage
change in stock price over any other disjoint time interval. Condition (2) means that
there are no jumps in the stock price, such as there would be in a Poisson process.
Condition (3) means that the probability of a 1% (or 12%) increase in the stock price
over one day is the same as the probabiity of a 1% (or 12%) increase over any other
day, and that the probability of a 1% (or 12%) increase over one hour is the same as
the probability of a 1% (or 12%) increase over any other other hour. Condition (4)
says that knowing the percentage change in any one day is of no help in predicting the
1
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View Full Documentpercentage change over any other day. One describes these conditions by saying that
the stock follows a geometric Brownian motion, or that the log of the price follows a
random walk.
It can be shown that if the market is con
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 '09
 GEANAKOPLOS,JOHN
 Mathematical finance, Black–Scholes

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